For-profit colleges have long been the target of government ire, starting with then-Education Secretary William J. Bennett’s criticism of their “shameful and tragic” financial-aid abuses in the 1980s and marked most recently by President Obama’s proposal to close a loophole that allows for-profit colleges to cash in on GI Bill benefits.

These institutions—places like Corinthian Colleges andAshford University—have generally come to be known for using deceptive and profit-maximizing tactics to recruit vulnerable populations paying for college with federal financial aid, delivering low-quality education, and saddling students with unmanageable amounts of debt.

But as a report released Tuesday by Robert Shireman at the Century Foundation points out, a “nonprofit” label doesn’t guarantee that your tuition is going toward an ethical or high-quality institution, either. In fact, you could be dealing with what he calls a “covert for-profit”—a former for-profit college that has converted to nonprofit status to avoid the growing number of regulations, while continuing to function like a for-profit institution.

Using case studies of four former for-profits that have converted to nonprofit status in the last two decades—Herzing University, Remington College, Everglades College, and the Center for Excellence in Higher Education—Shireman’s report suggests that the IRS may have overlooked the failure of these institutions to alter their governance and operations to match the requirements for gaining tax-exempt, nonprofit status. He points to potential conflicts of interest and other ways of operating that are “not at all consistent with what the organizations asserted when they were seeking the initial IRS approval.” The practice is particularly troubling in light of the fact that numerous other for-profit colleges, including Grand Canyon University, are vying for non-profit conversions.

Perhaps more important, Shireman’s report throws into question the myriad daily assumptions Americans make about nonprofit schools—and the resulting trust we place in them.

In a landscape of higher education that has come to include “public benefit” corporations (a classification somewhere between profit and non-profit, and allegedly geared toward “social good”), nonprofit institutions that pay officials exorbitant salaries, and others whose strategic plans seek to “maximize net tuition revenue,” the task of defining a “covert for-profit” college is hardly cut-and-dried.

The report certainly highlights a need for closer attention to and better regulation of these for-profit-to-nonprofit conversions by the IRS and the U.S. Department of Education. But as Mark S. Schneider of the American Institutes for Research (and a consultant toMONEY’s Best Colleges) told The Chronicle of Higher Education, it should also serve as a call for us to more closely examine all of our educational institutions, irrespective of labels.

“What we should be thinking about is how well any program—regardless of tax status, regardless of parent organization—how well they’re serving their students,” Schneider said.